•finding the Value of an infinite Series , <kc. 2.07 
Sec. in their natural order; to which if we add an unit, 
the numbers thereby produced will be the even numbers 
2, 4, 6, 8, 10, 12, 14, 16, &c. in their natural order, 
which are the doubles of the natural numbers, 1, 2, 3, 
4 ) 5 ) 7, &c. Therefore the number of terms of that 
feries from the beginning of it to any given term in it, 
including the faid term, is always half the number that 
is produced by adding an unit to the index of t in the 
faid term. Thus, if we take the term 777^, and add 1 to 
11, which is the index of the power of t in it, the fum 
will be 1 2, the half of which is 6, which is the number 
of terms in the feries from the beginning of it to the 
t 11 
term 777-0, including the faid term, that term being the 
fixth term in the feries. If therefore we take the term 
t 999 # 
— 7 - ,. ', and are defirous of knowing its place in the fe- 
ries, or the number of terms from the beginning of the 
feries to that term inclufively, we muft add 1 to the in- 
dex of the power of t in its numerator, which will in- 
creafe it to 1000 ; and half this fum, to wit, 500, will 
be the number of terms from the beginning of the feries 
t 999 
to the term - 99ii - inclufively ; or, in other words, this 
term w r ill be the 500th term of the feries. To arrive 
therefore at thofe terms of the feries in which the in- 
dexes 
